I see these examples (not just for the bard, but for all classes) as very good ways of showing how those classes exist in the world, what their roles are, and how they fit into the game from a narrative perspective.
That being said, the examples need not represent spellcasting at all. Not all spellcasting requires material components or a focus of some sort. In the examples provided, I can just as easily see those as representations of the bard's class features rather than the bard casting spells. I also don't know that the examples need to exactly match what can be done in game, for the same reason -- they're narrative examples, not gameplay examples. They might also be incomplete examples.
The first example is most likely a demonstration of the bard casting Legend Lore, which has V, S, and M components. The M components for this spell have a specific cost (250gp worth of incense and 4 ivory strips worth at least 50gp each), which means a bardic focus can't be used to cast it, so in this case, if she is casting Legend Lore (and it sounds like it, from the narrative description) there is no bardic focus.
The second and third examples seem to be narrative descriptions of various uses of the bardic class feature Bardic Inspiration which also doesn't require any sort of focus or musical instrument.
To address your specific concerns:
Is it reasonable to assume that RAW or at least RAI that the
implication is that in the case of bards, they can use 1) their voice,
2) an improvised musical instrument, or 3) a bought musical
instrument?
As far as RAW is concerned, you have one option: a musical instrument. I understand this is vague. The description in Chapter 5, Equipment (5e SRD) says:
Musical Instrument. Several of the most common types of musical instruments are shown on the table as examples. If you have proficiency with a given musical instrument, you can add your proficiency bonus to any ability checks you make to play music
with the instrument. A bard can use a musical instrument as a spellcasting focus. Each type of musical instrument requires a separate proficiency.
The table in question is the Tools table, relevant section reproduced below:
\begin{array}{r|ccc}
\text{Musical Instrument} & \text{Cost(gp)} & \text{Weight(lbs)}\\
\hline
\text{Bagpipes} & 30 & 6 \\
\text{Drum} & 6 & 3 \\
\text{Dulcimer} & 25 & 10 \\
\text{Flute} & 2 & 1 \\
\text{Lute} & 35 & 2 \\
\text{Lyre} & 30 & 2 \\
\text{Horn} & 3 & 2 \\
\text{Panflute} & 12 & 2 \\
\text{Shawm} & 2 & 1 \\
\text{Viol} & 30 & 1 \\
\end{array}
This list is not meant to be exhaustive, of course, and your bard could use any musical instrument that exists in your world. However, the fact that playing a musical instrument has a specific entry in the rules says to me it's more than just banging on something and making noise. The fact the they require proficiency says to me that not just anyone can bang on a drum to make music. I also take it to mean that the bard requires proficiency in the instrument in order to use it as a bardic focus. This is not explicit but it stands to reason, since you need proficiency to 'use' a musical instrument, and a bard must use the instrument as a bardic focus -- this is the RAI part. I have a hard time believing that any designer intended to allow a bard to use a lute as a focus without being able to actually play the lute.
Taking all of this into consideration, I think it would be a stretch to allow a bard to bang on his armor, hum a few bars, or use some other type of improvised musical instrument, and without proficiency, use that as his bardic spellcasting focus.
Voice -- No. And even if so, you'd have a hard time saying your
verbal components if you're also humming out a few bars as your
arcane focus.
Improvised instrument -- No. You don't have
proficiency in "banging on your armor."
A purchased instrument -- Yes. No problem there; it's in the rules.
And if such an assumption isn't RAW/RAI, what are the implications to
allowing it as a house rule?
Not too much of a problem here. It's a cool narrative device. Spellcasting foci aren't required ("you can use a [thing] as a spellcasting focus" in every mention of foci in the class rules). It doesn't really affect the bard's ability to cast his spells whether he has a focus or not -- there's always the option of a component pouch -- and not all spells require foci. All in all if you want to allow it there's no real problem; in the end it doesn't make much of a difference.
Best Answer
RAW, switching either costs an action, or requires two free object interactions (this requiring two of the same action in a row)
You are allowed one free object interaction on your turn (PHB pg 190), as a part of your action or movement (implied that the free interaction must be related to that, such as opening a door or drawing a weapon). Any more than that requires an action.
So you cannot put away your instrument and draw your blade as free actions on the same turn. That is two object interactions. (Compare this to the Dual Wielder feat, which explicitly allows you to draw or stow two weapons on the same turn)
The way to switch without taking an action is best explained with an example combat:
So RAW, you have to effectively take at least two of the same type of action (spellcasting or weapon attack) before switching.
You could drop one of the items (weapon or instrument), and use the interaction to get to the other weapon in a single turn, but that requires going back and picking it up later when you want to be able to use it again (RAW, this will require an action, unless part of an attack or spellcast, depending on the item being picked up).
You could tie the instrument to your body (letting you drop it without consuming your free object interaction) or use a component pouch, eliminating the need for the object interaction on round 4, but not the interaction required on rounds 2 or 6. This would allow a combat to go weapon, spell, weapon, weapon, spell, weapon.
However, many DMs will find this needlessly pedantic, hard to keep track of, and generally un-fun, and will let you draw and stow (or stow and draw) on the same turn.
Remember that all of this assumes that you need a hand free for the Somatic/Material components. Bards have lots of Verbal only spells, and you may not need to drop/stow your weapon to be able to cast things like Vicious mockery, dissonant whispers, faerie fire, and blindness/deafness. You will still need to follow the draw/stow rules for spells with somatic components, however.
The War Caster feat also interacts with this, as it allows you to use your shield or weapon hand to preform the somatic requirements for a spell, meaning that if your spell does not have material components, then you do not need to pull out your focus. This unlocks many other spells, such as Cure Wounds, Silence, Dispel Magic, Greater Invisibility, Dominate Person, Eyebite, etc...